1.

Solve the following equation : `3^(x+2)+3^(-x)=10`

Answer» Given equation is
`3^(x+2)+3^(-x)=10`
`implies3^(x)xx3^(2)+(1)/(3^(x))=10implies9xx3^(x)+(1)/(3^(x))=10" ".....(2)`
Let `3^(x)=a``3^(x)=a" "......(2)`
Then from (1)
`9a+(1)/(a)=10`
`implies9a^(2)+1=10a " "implies9a^(2)-10a+1=0`
`implies9a^(2)-(9+1)a+1=0" "implies9a^(2)-9a-a+1=0`
`implies9a(a-1)-1(a-1)=0" "implies(9a-1)(a-1)=0`
`implies9a-1=0" "or" "a-1=0`
when `9a-1=0impliesa=(1)/(9)`
and when `a-1=0impliesa=1`
Substituting values of a in equation (2)
when `a=(1)/(9)`
`3^(x)=(1)/(9)" "implies" "3^(x)=(1)/(3^(2)`
`implies3^(x)=3^(-2)" "or" "x=-2`
when a=1
`3^(x)=1`
`implies3^(x)=3^(0)" "implies" "x=0`
Hence, 0 and -2 are roots of the equation.


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